The Evergreen Project: How To Learn From Mistakes Caused by Blurry Vision in MAX-CSP Solving
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Transcript of The Evergreen Project: How To Learn From Mistakes Caused by Blurry Vision in MAX-CSP Solving
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PhD March 2007 1
The Evergreen Project:How To Learn From Mistakes Caused by Blurry Vision in
MAX-CSP Solving
Karl J. Lieberherr
Northeastern University
Boston
joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart
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PhD March 2007 2
Where we are
• Introduction
• Look-forward
• Look-backward
• Packed truth tables
• SPOT: how to use the look-ahead polynomials (look-forward) together with superresolution (look-backward).
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Problem Snapshot• SAT: classic problem in complexity theory• SAT & MAX-SAT Solvers: working on CNFs (a
multi-set of disjunctions).
• CSP: constraint satisfaction problem– Each constraint uses a Boolean relation.– e.g. a Boolean relation 1in3(x y z) is satisfied iff
exactly one of its parameters is true.
• CSP & MAX-CSP Solvers: working on CSP instances (a multi-set of constraints).
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Introduction
• Boolean MAX-CSP(G) for rank d, G = set of relations of rank d– Input
• Input = Bag of Constraint• Constraint = Relation + Set of Variable• Relation = int. // Relation number < 2 ^ (2 ^ d) in G• Variable = int
– Output• (0,1) assignment to variables which maximizes the number of
satisfied constraints.
• Example Input: G = {22} of rank 3– 22:1 2 3 0 – 22:1 2 4 0 – 22:1 3 4 0 1in3 has number 22
M = {1 !2 !3 !4} satisfies all
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Variation
MAX-CSP(G,f): Given a MAX-CSP(G) instance expressed in n variables
which may assume only the values 0 or 1, find an assignment to the n variables which satisfies at least the fraction f of the constraints.
Example: G = {22} of rank 3MAX-CSP({22},f):
22:1 2 3 0 22:1 2 4 0 22:1 3 4 022: 2 3 4 0
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Our Approach
• Superresolution & P-Optimality Based MAX-CSP Solver
• Highlights– Look Forward (in Abstract Representation)– Look Backward (in Transition System)– Packed Truth Tables (in Intermediate Representation)
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Where we are
• Introduction
• Look-forward
• Look-backward
• Packed truth tables
• SPOT: how to use the look-ahead polynomials together with superresolution.
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Look Forward
• Why?– To make informed decisions
• How?– Abstract representation based on look-ahead
polynomials
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Look-ahead Polynomial(Intuition)
• The look-ahead polynomial computes the expected fraction of satisfied constraints among all random assignments that are produced with bias p.
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Consider an instance: 40 variables,1000 constraints (1in3)
1, … ,40
22: 6 7 9 0
22: 12 27 38 0
Abstract representation:reduce the instance tolook-ahead poly. 3p(1-p)2
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PhD March 2007 11
1in3
0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Coin bias (Probability of setting a variable to true)
Fra
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3p(1-p)2 for MAX-CSP({22})
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Look-ahead Polynomial(Definition)
• F is a MAX-CSP(G) instance.
• N is an arbitrary assignment.
• The look-ahead polynomial laF,N(p) computes the expected fraction of satisfied constraints of F when each variable in N is flipped with probability p.
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The general case MAX-CSP(G)
G = {R1, … }, tR(F) = fraction of constraints in F that use R.
x = p
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Look-ahead Polynomial in Action
• Focus on purely mathematical question first
• Algorithmic solution will follow
• Mathematical question: Given a MAX-CSP(G) instance. For which fractions f is there always an assignment satisfying fraction f of the constraints? In which constraint systems is it impossible to satisfy many constraints?
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Remember?
MAX-CSP(G,f): Given a MAX-CSP(G) instance expressed in n variables
which may assume only the values 0 or 1, find an assignment to the n variables which satisfies at least the fraction f of the constraints.
Example: G = {22} of rank 3MAX-CSP({22},f):
22:1 2 3 0 22:1 2 4 0 22:1 3 4 022: 2 3 4 0
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Simple example
MAX-CSP({22},f):
For f <= u: problem has always a solutionFor f = u + : problem has not always a solution,
u critical transition point
always (fluid)
not always (solid)
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The Magic Number
• u = 4/9
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1in3
0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Coin bias (Probability of setting a variable to true)
Fra
ctio
n o
f co
nst
rain
ts t
hat
ar
e g
uar
ante
ed t
o b
e sa
tisf
ied
3p(1-p)2 for MAX-CSP({22})
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Produce the Magic Number
• Use an optimally biased coin– 1/3 in this case
• In general: min max problem
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General Dichotomy Theorem
MAX-CSP(G,f): For each finite set G of relationsthere exists an algebraic number tG
For f <= tG: MAX-CSP(G,f) has polynomial solutionFor f = tG+ : MAX-CSP(G,f) is NP-complete,
tG critical transition point
easy (fluid)
hard (solid)
due to Lieberherr/Specker
polynomial solution:Use optimally biased coin.Derandomize.P-Optimal.
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Observations
• The look-ahead polynomial look-forward approach has not been used in state-of-the-art MAX-SAT and Boolean MAX-CSP solvers.
• Often a fair coin is used. The optimally biased coin is often significantly better.
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PhD March 2007 24N0 ={!v1,!v2,!v3,!v4}
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PhD March 2007 25N0‘ ={v1,!v2,!v3,!v4}
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SAT Rank 2 example9 constraints
14 : 1 2 014 : 3 4 014 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0
14: 1 2 = or(1 2) 7: 1 3 = or(!1 !3)
What is the look-aheadpolynomial?
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PhD March 2007 27appmean = lookahead is an approximation of the true mean
Blurry vision
What do we learn from the abstract representation?• set 1/3 of the variables to true (maximize).• the best assignment will satisfy at least 7/9 constraints.• very useful but the vision is blurred in the “middle”.
excellent peripheral vision
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PhD March 2007 28
Where we are
• Introduction
• Look-forward
• Look-back
• Packed truth tables
• SPOT: how to use the look-ahead polynomials
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Look Backward
• Why?– to avoid past mistakes
• How?– Transition system based on superresolution
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Observation
• Optimally biased coin technique based on look-ahead polynomials is “best-possible”.
• If we could improve it by a trillionth in polynomial time, then P=NP.
• We improve it now by learning new constraints that will influence the polynomial.
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Clause Learning• Let’s go beyond what an optimally biased
coin guarantees!• Goal: satisfy the maximum number of
constraints. • Approach: Superresolution.
– When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease
• A mistake is made.
– Who to blame: the decision literals• They are the culprits.
– How to penalize: add the disjunctions of their negations as a superresolvent
• The gang of culprits is watched.
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Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where k Md
M || F || SR || N → M || F || SR, NewSR || N
• if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).
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Algorithm plan
• start with an arbitrary assignment N.
• while (proof incomplete) {– try to improve N by creating new assignment
from scratch using optimally biased coin to flip the assignments;
• success: Update N;• failure: learn a new constraint that will prevent
same mistake and will “improve” the polynomial. }
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PhD March 2007 36
Properties of TS
• TS finds the maximum in a finite number of steps.
• It creates a proof that we indeed found the maximum.
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PhD March 2007 37
Optimized Semi-Superresolution
• Not all decision literals may be responsible for the “mistake”.
• Want to find a minimal superresolvent so that deleting one literal would destroy the superresolvent property.
• Can be implemented by a traversal back the implication graph that is built as part of unit propagation.
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PhD March 2007 38
Where we are
• Introduction
• Look-forward
• Look-back
• Packed Truth Tables
• SPOT: how to use the look-ahead polynomials
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PhD March 2007 39
Requirements for Packed Truth Tables
• The look-ahead polynomial can be computed efficiently. Requires efficient truth table analysis.
• Reduction of an instance must be efficient.
• Efficiently compute the forced variables.
• Each relation has a unique representation.
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Packed Truth Tables
22 254
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RelationI: implemented by bitwise operations
int isForced(int variablePosition)boolean isIrrelevant(int variablePosition)int nMap(int variablePosition)int numberOfRelevantVariables()int q(int s) int reduce(int variablePosition, int value)int rename(int permutationSemantics,
int... permutation)
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PhD March 2007 42
Where we are
• Introduction
• Look-forward
• Look-back
• Packed truth tables
• SPOT: how to use the look-ahead polynomials with superresolution
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PhD March 2007 43
Using the look-ahead polynomials
• Value Ordering– Decide: how to set the variable
• Variable Ordering– Which variable to set next
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PhD March 2007 44
There is hope that the look-ahead polynomials are useful
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PhD March 2007 45
What is new?
• New: Packed Truth Tables
• New: Superresolution for MAX-CSP
• New: Integration of look-ahead polynomials with superresolution
• Old: Superresolution for SAT (1977)
• Old: Look-ahead polynomials (1983)
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PhD March 2007 46
Future work
• Exploring best combination of look-forward and look-back techniques.
• Find all maximum-assignments or estimate their number.
• Robustness of maximum assignments.
• Are our MAX-CSP solvers useful for reasoning about biological pathways?
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PhD March 2007 47
Conclusions
• Presented SPOT, a family of MAX-CSP solvers based on look-ahead polynomials and non-chronological backtracking.
• SPOT has a desirable property: P-optimal.
• Preliminary experimental results are encouraging.
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PhD March 2007 48
end for now
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PhD March 2007 49
Rank 2 example
• 14 : 1 2 014 : 3 4 014 : 5 6 0 7 : 1 3 0 7 : 1 5 0 7 : 3 5 0 7 : 2 4 0 7 : 2 6 0 7 : 4 6 0
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PhD March 2007 50appmean is an approximation of the true mean
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Transition Manager
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MAX-CSP:Superresolution and P-Optimality
Karl J. Lieberherr
Northeastern University
Boston
joint work with Ahmed Abdelmeged, Christine Hang and Daniel Rinehart
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Binomial Distribution
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Example
x1 + x2 + x3 = 1x1 + x2 + + x4 = 1 can satisfy 6/7x1 + x3 + x4 = 1 x1 + x3 + x4 = 1x1 + x2 + + x5 = 1x1 + x3 + x5 = 1 x2 + x3 + x5 =1
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PhD March 2007 59
1in3
0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Coin bias (Probability of setting a variable to true)
Fra
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ied
maximize 3x(1-x)2
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Organization of Solver
look back look forward
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Transition Rules
• Unit-Propagation (UP):
M || F || SR || N → Mk || F || SR || N
• if k is undefined in M, and• unsat (M¬k,SR) > 0 or unsat(M¬k,F) ≥ unsat(N,F).
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Transition Rules
• Decide (D):
M || F || SR || N → Mkd || F || SR || N
• if k is undefined in M, and• v(k) occurs in some constraint of F.
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Transition Rules
• Update:
M || F || SR || N → M || F || SR || M
• if M is complete, and• unsat(M,F) < unsat(N,F).
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Transition Rules
• Restart:
M || F || SR || N → { } || F || SR || N
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Transition Rules
• Finale:
M || F || SR || N → M || F || SR || N
• if Φ SR or unsat(N,F) = 0.
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Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where k Md
M || F || SR || N → M || F || SR, NewSR || N
• if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).
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Transition Rules
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Transition Rules (cont.)
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Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where k Md
M || F || SR || N → M || F || SR, NewSR || N
• if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).
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Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where k Md
M || F || SR || N → M || F || SR, NewSR || N
• if unsat(M,SR) > 0 or unsat(M,F) ≥ unsat(N,F).
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Transition Rules
• Semi-Superresolution (SSR):
NewSR = V (¬k), where kєM’ subset Md
M || F || SR || N → M || F || SR, NewSR || N
• if mistake(M) and UP*(reduce(F,A(NewSR)))
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Our Approach
• Superresolution & P-Optimality Based MAX-CSP Solver
• Highlights– Optimally Biased Coin (in Abstract Representation)– Clause Learning (in Transition System)– Bitwise Relation Reduction (in Intermediate
Representation)
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Clause Learning• Let’s go beyond what an optimally biased
coin guarantees!• Goal: satisfy the maximum number of
constraints. • Approach: Superresolution.
– When to apply: number of constraints guaranteed to be unsatisfied doesn’t decrease
• A mistake is made.
– Who to blame: the decision literals• They are the culprits.
– How to penalize: add the disjunctions of their negations as a superresolvent
• The gang of culprits is watched.
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Sudoku